Testing a Simple Recipe for Estimating Thermal Hydrodynamic Escape Rates in Primitive Terrestrial Atmospheres

Abstract:

During the first billion years of the Sun’s history, the emission of ultraviolet and X-ray radiation varied from ~100 to ~6 times greater than its present level. The absorption of this intense radiation in the upper atmospheres of the terrestrial planets is believed to have driven rapid hydrodynamic escape, either in the form of energy-limited escape or transonic blow-off. The calculation of escape rates under these circumstances, and in particular the nature of the correct condition to apply at the upper boundary, depends on whether or not the flow remains subsonic below the exobase. If the flow remains subsonic, the kinetic Jeans equations may be applied at the exobase; otherwise, the radius of the sonic point must be located and then appropriate boundary conditions applied at this radius. This seems to suggest that the full hydrodynamic escape problem needs to be solved iteratively to determine where the sonic radius falls and the type of boundary conditions that should be applied. Such an arduous undertaking is generally impractical for standard application in chemical evolution models or related studies. Fortunately, a much easier but still accurate approach to determining whether the flow remains subsonic below the exobase for a given amount of energy deposition has been provided by Johnson et al. (2013, Ap. J. Lett. 768:L4), who base their results on rigorous Discrete Simulation Monte Carlo models. Their model provides the ratio of the escape rate to the energy-limited value as a function of the total XUV heating. The XUV heating, however, is itself coupled to the escape rate through the radial structure of the upper atmosphere, which can become greatly distended for large heating rates. Here we present a simple recipe for estimating the hydrodynamic escape rate that includes the coupling between the escape rate, the radial structure, and the XUV heating while avoiding the use of demanding numerical calculations. The approach involves an iterative semi-analytical method for determining the effective radius of energy deposition, from which the escape rate, radial structure, and other parameters can be derived. We test its performance against some more elaborate, rigorous calculations of primitive-atmosphere hydrodynamic escape that are available in the literature.